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<title>Xah: Special Plane Curves: Astroid</title>
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<pre>back to <a href="../specialPlaneCurves.html">Table of Contents</a></pre>
<h1><a name="Top">Astroid</a></h1>

<center>
<img src="astroid.png" width="280" height="280"><br>
Astroid and its <a href="../Parallel_dir/parallel.html">parallels</a>
</center>


<p><img src="../Icons_dir/mmaIconSmall.gif" width="16" height="16"> <a href="astroid.nb">Mathematica Notebook for This Page</a>.</p>

<!-- page -->
<table border="1">
<tr>
<TD><a href="#History">History</a></TD>
<TD><a href="#Description">Description</a></TD>
<TD><a href="#Formulas">Formulas</a></TD>
<TD><a href="#Properties">Properties</a></TD>
<TD><a href="#Related%20Web%20Sites">Related Web Sites</a></TD>
</tr>
</table>

<hr>
<h2><a name="History">History</a></h2>

<p>The cycloidal curves, including the astroid, were discovered by Roemer (1674) in his search for the best form for gear teeth. Double generation was first noticed by Daniel Bernoulli in 1725. [verbatim, Robert C. Yates, 1952]</P>

<p>The astroid seems to have acquired its present name only in 1838, in a book published in Vienna; it went, even after that time, under various other names, such as cubocycloid, paracycle, four-cusp-curve, and so on. The equation x^(2/3) + y^(2/3) == a^(2/3) can, however, be found in Leibniz's correspondence as early as 1715. [verbatim, E.H.Lockwood, 1961]</P>

<hr><h2><a name="Description">Description</a></h2>

<p>Astroid is a special case of <a href="../Hypotrochoid_dir/hypotrochoid.html">hypotrochoid</a>. (see <a href="../Intro_dir/familyIndex.html#Curve%20Family%20Tree">Curve Family Index</a>). Astroid is defined as the trace of a point on a circle of radius r rolling inside a fixed circle of radius 4 r or 4/3 r. The latter is known as double generation.</P>
<center>
<table>
<tr align=center>
<td> 
<img src="astroidGen1.png" width="200" height="200"><br>
<img src="../Icons_dir/movieIconSmall.gif" width="13" height="16">
<a href="astroidGen1.mov">Tracing Astroid</a><br>
<img src="../Icons_dir/gspIconSmall.gif" width="14" height="16">
<a href="astroidDoubleGen.gsp">astroid constructions</a><br>
</td>
<td> 
<img src="astroidGen2.png" width="200" height="200"><br>
<img src="../Icons_dir/movieIconSmall.gif" width="13" height="16">
<a href="astroidGen2.mov">Double Generation</a></td>
</tr>
</table>
</center>


<hr><h2><a name="Formulas">Formulas</a></h2>

<p>The following formulas describe an astroid centered on the origin, and the length from center to one cusp is a, where a is a scaling factor.</P>

<!-- xahnote: prove astroid eq from parametric: (x^2 +y^2 -1)^3 + 27 * x^2 * y^2 == 0 is equivalent to x^(2/3) + y^(2/3) == 1. And, in general, what are the exhaustive list of operations are allow on f[x]==0 to generate the same curve? where f is not necessarily polynomial. -->


<p>Parametric: {Cos[t]^3, Sin[t]^3}, 0 &lt; t ≤ 2 * π. <img src="../Icons_dir/gcf.gif" width="12" height="16"> <a href="astroid_para_plot.gcf">parametric plot</a></p>

<p>Cartesian: (x^2 +y^2 -1)^3 + 27 * x^2 * y^2 == 0. This equation is centered on origin and a cusp at {1,0}. Replace x by x/a and y by y/a and multiply both sides by a^6 and we obtain the classic equation given with scaling factor a as: (x^2 + y^2 - a^2)^3 + 27*a^2*x^2*y^2 == 0.</p>

<P>another equivalent equation is: x^(2/3) + y^(2/3) == 1. <img src="../Icons_dir/gcf.gif" width="12" height="16"> <a href="astroid_eq_plot.gcf">astroid_eq_plot.gcf</a></p>


<P><img src="../Icons_dir/mmaIconSmall.gif" width="16" height="16"> <a href="astroid_eq_proof.nb">derivation of different formulas</a></p>

<P>The following formulas are not verified:</p>
<ul>
<li>Pedal: r^2==a^2-3*p^2.</li>
<li>Area: 3/8*π*a^2.</li>
<li>Surface of Revolution: 12/5*π*a^2.</li>
<li>Volumn of Revolution: 32/105*π*a^3.</li>
<li>ϕ == π - t.</li>
</ul>


<hr>
<h2><a name="Properties">Properties</a></h2>

<h3><a name="Trammel of Archimedes and Envelope of Ellipses">Trammel of Archimedes and Envelope of Ellipses</a></h3>

<p>Define the axes of the astroid to be the two perpendicular lines passing its cusps. Property: The length of tangent cut by the axes is constant. A mechanical devise where a fixed bar with endings sliding on two penpendicular tracks is called the Trammel of Archimedes. The <a href="../Envelope_dir/envelope.html">envelope</a> of the moving bar is then the astroid. A fixed point on the bar will trace out an <a href="../Ellipse_dir/ellipse.html">ellipse</a>. (see left figure)</P>

<p>Astroid is also the <a href="../Envelope_dir/envelope.html">envelope</a> of co-axial <a href="../Ellipse_dir/ellipse.html">ellipses</a> whose sum of major and minor axes is contsant.(see right figure)</P>
<center>
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<img src="astroidTrammel.png" width="250" height="250"><br>
<img src="../Icons_dir/movieIconSmall.gif" width="13" height="16">
<a href="astroidTrammel.mov">Trammel of Archimedes</a><br>
<img src="../Icons_dir/gspIconSmall.gif" width="14" height="16">
<a href="astroidTrammel.gsp">Trammel of Archimedes</a><br>
<img src="../Icons_dir/cabriIconSmall.gif" width="14" height="16">
<a href="astroidTrammel.fig">Trammel of Archimedes</a>
</td>
<td valign=top>
<img src="astroidByEllipse.png" width="250" height="250"><br>
<img src="../Icons_dir/movieIconSmall.gif" width="13" height="16">
<a href="astroidByEllipse.mov">Envelope of Ellipses.</a>
</td>
</tr>
</table>
</center>

<h3>Evolute of Astroid</h3>

<p>The <a href="../Evolute_dir/evolute.html">evolute</a> of an astroid is another astroid. (all <a href="../EpiHypocycloid_dir/epiHypocycloid.html">epi/hypocycloids</a>' evolute is equal to themselves) In the figure on the left, points on the curve are connected to their center of osculating circles. On the right, the <a href="../Evolute_dir/evolute.html">evolute</a> is drawn as the <a href="../Envelope_dir/envelope.html">envelope</a> of normals.</P>
<center>
<table>
<tr align=center>
<td><img src="astroidEvolute.png" width="250" height="250"></td>
<td><img src="astroidEvoluteByNorm.png" width="250" height="250"></td>
</tr>
</table>
</center>



<h3>Curve Construction</h3>

<P>The astroid is rich in properties that one can construct the curve, its tangent, and center of osculating circle, and device other mechanical ways to generate the curve. Here we illustrate one construction of the curve, its tangent, normal, and center of osculating circle.</p>

<P>
Let there be a circle centered on B passing K. We will construct an astroid centered on B with one cusp at K.
Let B be the origin, and K be the ponit {1,0}
Let L be a point on Circle[B,BK]. Drop a line from L perpendicular to x-axis, let M be their intersection. Similarly drop a line from L perpendicular to y-axis, call the intersection N.
Let P be a point on MN such that LP and MN are perpendicular.
Now, P is a point on the astroid, and MN is its tangent, LP is its normal.
Let D be the intersection of LP and circle[B,BK]. Let D' be the reflection of D thru MN.
Now, D' is the center of osculating circle at P.
</p>


<P>
<IMG SRC="astroidConstruction.png" ALT="" WIDTH="423" HEIGHT="409"><br>
<img src="../Icons_dir/gspIconSmall.gif" width="14" height="16"> <a href="astroid_const.gsp">astroid construction</a>
</P>

<h3>Pedal, Radial, and Rose</h3>

<p>The <a href="../Pedal_dir/pedal.html">pedal</a> of an astroid with respect to its center is 4 petalled <a href="../Rose_dir/rose.html">rose</a>, called a quadrifolium. Astroid's radial is also quadrifolium. (all <a href="../EpiHypocycloid_dir/epiHypocycloid.html">epi/hypocycloid</a>'s <a href="../Pedal_dir/pedal.html">pedal</a> and radial are equal, and they are <a href="../Rose_dir/rose.html">roses</a>.)</P>
<center>
<table>
<tr align=center>
<td>
<img src="astroidPedal.png" width="250" height="250"><br>
<img src="../Icons_dir/gspIconSmall.gif" width="14" height="16">
<a href="astroidPedal.gsp">Astroid's Pedal Curve</a>
</td>
<td><img src="astroidRadial1.png" width="250" height="250"></td>
</tr>
</table>
</center>



<h3>Deltoid and Astroid</h3>

<!-- xahnote: prove astroid's catacaustic is deltoid -->
<p>Astroid is the <a href="../Caustics_dir/caustics.html">catacaustic</a> of <a href="../Deltoid_dir/deltoid.html">deltoid</a> with parallel rays in any direction.</P>
<center>
<table>
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<td><img src="../Deltoid_dir/deltoidCaustic1.png" width="310" height="310"></td>
<td><img src="../Deltoid_dir/deltoidCaustic2.png" width="310" height="310"></td>
</tr>
</table><br>
<img src="../Icons_dir/movieIconSmall.gif" width="13" height="16"> <a href="../Deltoid_dir/deltoidCaustic.mov">Moving Light Source (234 k)</a>
</center>


<h3>Orthoptic</h3>
<!-- xahnote: prove astroid's orthoptic -->
<p>The orthoptic with respect to its center is r^2 == (1/2)*Cos[2*θ]^2. [Robert C. Yates.] Recall that an orthoptic with respet to a curve C and a point O is the locus of points P on all tangents T of C, such that OPT is a right angle.</p>

<img src="astroid_orthoptic.png" width="345" height="344">
<P><img src="../Icons_dir/gcf.gif" width="12" height="16"> <a href="astroid_othoptic.gcf">astroid_othoptic.gcf</a></p>



<hr>
<h2><a name="Related Web Sites">Related Web Sites</a></h2>

<p>see <b><A HREF="../Intro_dir/relatedHyperLinks.html">Generic Reference Page</A></b></P>

<p><a href="http://www-groups.dcs.st-andrews.ac.uk/%7Ehistory/Curves/Astroid.html">MacTutor Famous Curve Index</a></P>


<pre>back to <a href="../specialPlaneCurves.html">Table of Contents</a></pre>
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<pre>last updated: 2004-11.
&copy; copyright 1995-2004 by <a href="http://xahlee.org/PageTwo_dir/more.html">Xah Lee</a>. (<a href="mailto:xah@xahlee.org">xah@xahlee.org</A>)
http://xahlee.org/SpecialPlaneCurves_dir/specialPlaneCurves.html</pre>



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